↑

Properties

Label 1-517-517.53-r0-0-0
Degree $1$
Conductor $517$
Sign $0.561 + 0.827i$
Analytic cond. $2.40093$
Root an. cond. $2.40093$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.981 + 0.190i)2-s + (−0.435 + 0.900i)3-s + (0.927 − 0.373i)4-s + (0.149 + 0.988i)5-s + (0.256 − 0.966i)6-s + (−0.662 − 0.749i)7-s + (−0.839 + 0.542i)8-s + (−0.620 − 0.784i)9-s + (−0.334 − 0.942i)10-s + (−0.0682 + 0.997i)12-s + (−0.385 − 0.922i)13-s + (0.792 + 0.609i)14-s + (−0.955 − 0.295i)15-s + (0.721 − 0.692i)16-s + (−0.740 − 0.672i)17-s + (0.758 + 0.652i)18-s + ⋯
L(s)  = 1  + (−0.981 + 0.190i)2-s + (−0.435 + 0.900i)3-s + (0.927 − 0.373i)4-s + (0.149 + 0.988i)5-s + (0.256 − 0.966i)6-s + (−0.662 − 0.749i)7-s + (−0.839 + 0.542i)8-s + (−0.620 − 0.784i)9-s + (−0.334 − 0.942i)10-s + (−0.0682 + 0.997i)12-s + (−0.385 − 0.922i)13-s + (0.792 + 0.609i)14-s + (−0.955 − 0.295i)15-s + (0.721 − 0.692i)16-s + (−0.740 − 0.672i)17-s + (0.758 + 0.652i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.561 + 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.561 + 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(517\)    =    \(11 \cdot 47\)
Sign: $0.561 + 0.827i$
Analytic conductor: \(2.40093\)
Root analytic conductor: \(2.40093\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{517} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 517,\ (0:\ ),\ 0.561 + 0.827i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5569689098 + 0.2950867221i\)
\(L(\frac12)\) \(\approx\) \(0.5569689098 + 0.2950867221i\)
\(L(1)\) \(\approx\) \(0.5514196073 + 0.2044219536i\)
\(L(1)\) \(\approx\) \(0.5514196073 + 0.2044219536i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
47 \( 1 \)
good2 \( 1 + (-0.981 + 0.190i)T \)
3 \( 1 + (-0.435 + 0.900i)T \)
5 \( 1 + (0.149 + 0.988i)T \)
7 \( 1 + (-0.662 - 0.749i)T \)
13 \( 1 + (-0.385 - 0.922i)T \)
17 \( 1 + (-0.740 - 0.672i)T \)
19 \( 1 + (0.986 - 0.163i)T \)
23 \( 1 + (0.682 + 0.730i)T \)
29 \( 1 + (0.758 + 0.652i)T \)
31 \( 1 + (0.881 + 0.472i)T \)
37 \( 1 + (0.792 - 0.609i)T \)
41 \( 1 + (0.360 + 0.932i)T \)
43 \( 1 + (-0.0682 - 0.997i)T \)
53 \( 1 + (0.998 - 0.0546i)T \)
59 \( 1 + (-0.969 + 0.243i)T \)
61 \( 1 + (-0.282 - 0.959i)T \)
67 \( 1 + (-0.917 - 0.398i)T \)
71 \( 1 + (-0.981 - 0.190i)T \)
73 \( 1 + (0.554 + 0.832i)T \)
79 \( 1 + (0.946 - 0.321i)T \)
83 \( 1 + (0.410 + 0.911i)T \)
89 \( 1 + (0.460 - 0.887i)T \)
97 \( 1 + (0.721 + 0.692i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.75560905032559362513554482847, −22.52487041638170578484623658005, −21.63603551470662351153858738489, −20.735551352852949690519282236206, −19.64030382950187253134317383822, −19.262836329564461801916365013617, −18.347519621241262530916081206921, −17.56277423615553033694103285063, −16.73837168517069070504213528156, −16.21455355606459113126228033955, −15.16916100482950967063010173533, −13.64035200672890506369328037510, −12.792977614259727320213899594534, −12.05225713269572012153108740631, −11.5325522658850810290877940965, −10.20651179653347719799101379678, −9.21089375373795607253987611887, −8.586561574190218681698169310819, −7.6277708439275129380196670407, −6.513293645285132120596891490303, −5.92173619629981979835557806961, −4.56915120925548235696037709493, −2.77975009221864589128178731489, −1.91087061738480997823164632660, −0.78045759799213452814711055548, 0.763955582110628059322901945589, 2.76142215685865934339824572093, 3.35951793335021193209878524126, 4.9506540636517101977640143597, 6.04712494480174480002576033974, 6.89221193281532789434242294044, 7.64728600303665147772428664428, 9.1070633585023761739510871873, 9.816153483046025019117858457942, 10.48591227583369907973847809770, 11.10199943506873506333051833710, 12.04258143782992831261321487790, 13.57338684936602681582991918912, 14.58813938804388873376431214795, 15.50292585643243450940252685890, 15.98086567492249952953326691907, 17.00977711682703235320433547501, 17.71958701579178924031983335567, 18.32127316617330042564636083598, 19.68497162067888618209544405039, 20.001077043671696742803042213373, 21.11754158228541993670115975574, 22.06061121083120847412559326914, 22.850730373348984182516986331203, 23.46198767804150441181187006564

Graph of the $Z$-function along the critical line